Engineering Computations One:
Diﬀerential and Integral Calculus
Due Thursday 5 April 2012
Please ensure that all your working is clearly set out, all pages are stapled
together, and that your name along with the course name appears on the front
page of the assignment. Please place completed assignments in the assignment
box in WT level 1 by 5pm on the due date.
1. Evaluate each of the following limits or explain why it does not exist:
2. The current I at time t seconds in a series circuit containing only a resistor
with resistance 10 ohm, an inductor with inductance 0.5henry,anda
steady 12 volt battery connected at time t = 0 is given by the formula
1 − e
(this is shown on page 84 of the course manual). Brieﬂy
explain what happens to the current for t ≥ 0.
3. Sketch the graph of a function that satisﬁes all of the given conditions:
(−1) = 0, f
(1) does not exist, f
(x) < 0if|x| < 1, f
(x) > 0if
|x| > 1, f(−1) = 4, f(1) = 0, f
(x) < 0ifx =1.
(b) Domain g =(0, ∞), lim
+ g(x)=−∞, lim
(3) = 0, g
(x) < 0if1<x<3, g
(x) < 0ifx<2orx>4,
(x) > 0if2<x<4.
4. At what point(s) on the curve y =2(x − cos x) is the tangent parallel to
the line 3x−y = 5? Illustrate by drawing a rough sketch of the curve, the
line, and the tangent(s).
5. Suppose that f and g are diﬀerentiable functions and that F is the function
given by F(x)=f(x)g(x).
(a) Show that F
(b) Find similar formulas for F
6. Find when the function f(x)=
is increasing and decreasing.
7. Use calculus to determine whether the graph of f(x)=
upward or concave downward at x = π.
8. Suppose the position s of a particle at time t is given by s(t)=t tan t for
0 ≤ t ≤
(a) Find the velocity when t =
(b) Is the particle accelerating or decelerating when t =
The fact that
might be useful.
(d) Left-hand limit
Therefore, the limit does not exist.
Therefore, we have
The tangent is parallel to the line
The point is
By Leibnitz’ rule,
For critical points,
In the interval
In the interval
Therefore, the function is increasing on and decreasing on .
At the point
Therefore, the function is concave upwards.
(a) The velocity is given by
The velocity is
(b) The acceleration is given by
Therefore, the particle is accelerating.
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